Microbial Interaction Network Estimation via Bias-Corrected Graphical Lasso

With the increasing availability of microbiome 16S data, network estimation has become a useful approach to studying the interactions between microbial taxa. Network estimation on a set of variables is frequently explored using graphical models, in which the relationship between two variables is modeled via their conditional dependency given the other variables. Various methods for sparse inverse covariance estimation have been proposed to estimate graphical models in the high-dimensional setting, including graphical lasso. However, current methods do not address the compositional count nature of microbiome data, where abundances of microbial taxa are not directly measured, but are reflected by the observed counts in an error-prone manner. Adding to the challenge is that the sum of the counts within each sample, termed “sequencing depth,” is an experimental technicality that carries no biological information but can vary drastically across samples. To address these issues, we develop a new approach to network estimation, called BC-GLASSO (bias-corrected graphical lasso), which models the microbiome data using a logistic normal multinomial distribution with the sequencing depths explicitly incorporated, corrects the bias of the naive empirical covariance estimator arising from the heterogeneity in sequencing depths, and builds the inverse covariance estimator via graphical lasso. We demonstrate the advantage of BC-GLASSO over current approaches to microbial interaction network estimation under a variety of simulation scenarios. We also illustrate the efficacy of our method in an application to a human microbiome data set.

     

Gene Systems Network Inferred from Expression Profiles in Hepatocellular Carcinogenesis by Graphical Gaussian Model

Hepatocellular carcinoma (HCC) in a liver with advanced-stage chronic hepatitis C (CHC) is induced by hepatitis C virus, which chronically infects about 170 million people worldwide. To elucidate the associations between gene groups in hepatocellular carcinogenesis, we analyzed the profiles of the genes characteristically expressed in the CHC and HCC cell stages by a statistical method for inferring the network between gene systems based on the graphical Gaussian model. A systematic evaluation of the inferred network in terms of the biological knowledge revealed that the inferred network was strongly involved in the known gene-gene interactions with high significance Open image in new window , and that the clusters characterized by different cancer-related responses were associated with those of the gene groups related to metabolic pathways and morphological events. Although some relationships in the network remain to be interpreted, the analyses revealed a snapshot of the orchestrated expression of cancer-related groups and some pathways related with metabolisms and morphological events in hepatocellular carcinogenesis, and thus provide possible clues on the disease mechanism and insights that address the gap between molecular and clinical assessments.     

Equivariant Estimation for the Parameters of an Exponential Model Based on Censored Sampling

Lehmann (1983) provides a detailed discussion on equivariant estimation of the parameters of location, scale and location-scale models. Edwin Prabakaran and Chandrasekar (1994) developed simultaneous equivariant estimation approach and illustrated the method with examples. In this paper, we consider exponential models and obtain minimum risk equivariant estimators of the parameters based on Type II censored samples. The equivariant method of estimation is illustrated with biological and reliability applications.     

Estimation of a Joint Distribution Function with Multivariate Interval‐Censored Data when the Nonparametric MLE is not Unique

A nonparametric estimator of a joint distribution function F₀ of a d‐dimensional random vector with interval‐censored (IC) data is the generalized maximum likelihood estimator (GMLE), where d ≥ 2. The GMLE of F₀ with univariate IC data is uniquely defined at each follow‐up time. However, this is no longer true in general with multivariate IC data as demonstrated by a data set from an eye study. How to estimate the survival function and the covariance matrix of the estimator in such a case is a new practical issue in analyzing IC data. We propose a procedure in such a situation and apply it to the data set from the eye study. Our method always results in a GMLE with a nonsingular sample information matrix. We also give a theoretical justification for such a procedure. Extension of our procedure to Cox's regression model is also mentioned.     

Characterization of Microbial Dysbiosis and Metabolomic Changes in Dogs with Acute Diarrhea

Limited information is available regarding the metabolic consequences of intestinal dysbiosis in dogs with acute onset of diarrhea. The aim of this study was to evaluate the fecal microbiome, fecal concentrations of short-chain fatty acids (SCFAs), as well as serum and urine metabolites in healthy dogs (n=13) and dogs with acute diarrhea (n=13). The fecal microbiome, SCFAs, and serum/urine metabolite profiles were characterized by 454-pyrosequencing of the 16S rRNA genes, GC/MS, and untargeted and targeted metabolomics approach using UPLC/MS and HPLC/MS, respectively. Significantly lower bacterial diversity was observed in dogs with acute diarrhea in regards to species richness, chao1, and Shannon index (p=0.0218, 0.0176, and 0.0033; respectively). Dogs with acute diarrhea had significantly different microbial communities compared to healthy dogs (unweighted Unifrac distances, ANOSIM p=0.0040). While Bacteroidetes, Faecalibacterium, and an unclassified genus within Ruminococcaceae were underrepresented, the genus Clostridium was overrepresented in dogs with acute diarrhea. Concentrations of fecal propionic acid were significantly decreased in acute diarrhea (p=0.0033), and were correlated to a decrease in Faecalibacterium (ρ=0.6725, p=0.0332). The predicted functional gene content of the microbiome (PICRUSt) revealed overrepresentations of genes for transposase enzymes as well as methyl accepting chemotaxis proteins in acute diarrhea. Serum concentrations of kynurenic acid and urine concentrations of 2-methyl-1H-indole and 5-Methoxy-1H-indole-3-carbaldehyde were significantly decreased in acute diarrhea (p=0.0048, 0.0185, and 0.0330, respectively). These results demonstrate that the fecal dysbiosis present in acute diarrhea is associated with altered systemic metabolic states.     

Median Regression Model with Interval Censored Data

Quantile regression methods have been used to estimate upper and lower quantile reference curves as the function of several covariates. Especially, in survival analysis, median regression models to the right‐censored data are suggested with several assumptions. In this article, we consider a median regression model for interval‐censored data and construct an estimating equation based on weights derived from interval‐censored data. In a simulation study, the performances of the proposed method are evaluated for both symmetric and right‐skewed distributed failure times. A well‐known breast cancer data are analyzed to illustrate the proposed method.     

Identification of ligand binding sites of proteins using the Gaussian Network Model

The nonlocal nature of the protein-ligand binding problem is investigated via the Gaussian Network Model with which the residues lying along interaction pathways in a protein and the residues at the binding site are predicted. The predictions of the binding site residues are verified by using several benchmark systems where the topology of the unbound protein and the bound protein-ligand complex are known. Predictions are made on the unbound protein. Agreement of results with the bound complexes indicates that the information for binding resides in the unbound protein. Cliques that consist of three or more residues that are far apart along the primary structure but are in contact in the folded structure are shown to be important determinants of the binding problem. Comparison with known structures shows that the predictive capability of the method is significant.     

Automatic domain decomposition of proteins by a Gaussian Network Model

Proteins are often comprised of domains of apparently independent folding units. These domains can be defined in various ways, but one useful definition divides the protein into substructures that seem to move more or less independently. The same methods that allow fairly accurate calculation of motion can be used to help classify these substructures. We show how the Gaussian Network Model (GNM), commonly used for determining motion, can also be adapted to automatically classify domains in proteins. Parallels between this physical network model and graph theory implementation are apparent. The method is applied to a nonredundant set of 55 proteins, and the results are compared to the visual assignments by crystallographers. Apart from decomposing proteins into structural domains, the algorithm can generally be applied to any large macromolecular system to decompose it into motionally decoupled sub-systems.     

Graphical Aids to the Estimation and Discrimination of Uncertain Numerical Data

This research investigates the performance of graphical dot arrays designed to make discrimination of relative numerosity as effortless as possible at the same time as making absolute (quantitative) numerosity estimation as effortful as possible. Comparing regular, random, and hybrid (randomized regular) configurations of dots, the results indicate that both random and hybrid configurations reduce absolute numerosity estimation precision, when compared with regular dots arrays. However, discrimination of relative numerosity is significantly more accurate for hybrid dot arrays than for random dot arrays. Similarly, human subjects report significantly lower levels of subjective confidence in judgments when using hybrid dot configurations as compared with regular configurations; and significantly higher levels of subjective confidence as compared with random configurations. These results indicate that data graphics based on the hybrid, randomized-regular configurations of dots are well-suited to applications that require decisions to be based on numerical data in which the absolute quantities are less certain than the relative values. Examples of such applications include decision-making based on the outputs of empirically-based mathematical models, such as health-related policy decisions using data from predictive epidemiological models.     

iGNM: a database of protein functional motions based on Gaussian Network Model

MOTIVATION: The knowledge of protein structure is not sufficient for understanding and controlling its function. Function is a dynamic property. Although protein structural information has been rapidly accumulating in databases, little effort has been invested to date toward systematically characterizing protein dynamics. The recent success of analytical methods based on elastic network models, and in particular the Gaussian Network Model (GNM), permits us to perform a high-throughput analysis of the collective dynamics of proteins. RESULTS: We computed the GNM dynamics for 20 058 structures from the Protein Data Bank, and generated information on the equilibrium dynamics at the level of individual residues. The results are stored on a web-based system called iGNM and configured so as to permit the users to visualize or download the results through a standard web browser using a simple search engine. Static and animated images for describing the conformational mobility of proteins over a broad range of normal modes are accessible, along with an online calculation engine available for newly deposited structures. A case study of the dynamics of 20 non-homologous hydrolases is presented to illustrate the utility of the iGNM database for identifying key residues that control the cooperative motions and revealing the connection between collective dynamics and catalytic activity.     

Estimation of Thalamocortical and Intracortical Network Models from Joint Thalamic Single-Electrode and Cortical Laminar-Electrode Recordings in the Rat Barrel System

A new method is presented for extraction of population firing-rate models for both thalamocortical and intracortical signal transfer based on stimulus-evoked data from simultaneous thalamic single-electrode and cortical recordings using linear (laminar) multielectrodes in the rat barrel system. Time-dependent population firing rates for granular (layer 4), supragranular (layer 2/3), and infragranular (layer 5) populations in a barrel column and the thalamic population in the homologous barreloid are extracted from the high-frequency portion (multi-unit activity; MUA) of the recorded extracellular signals. These extracted firing rates are in turn used to identify population firing-rate models formulated as integral equations with exponentially decaying coupling kernels, allowing for straightforward transformation to the more common firing-rate formulation in terms of differential equations. Optimal model structures and model parameters are identified by minimizing the deviation between model firing rates and the experimentally extracted population firing rates. For the thalamocortical transfer, the experimental data favor a model with fast feedforward excitation from thalamus to the layer-4 laminar population combined with a slower inhibitory process due to feedforward and/or recurrent connections and mixed linear-parabolic activation functions. The extracted firing rates of the various cortical laminar populations are found to exhibit strong temporal correlations for the present experimental paradigm, and simple feedforward population firing-rate models combined with linear or mixed linear-parabolic activation function are found to provide excellent fits to the data. The identified thalamocortical and intracortical network models are thus found to be qualitatively very different. While the thalamocortical circuit is optimally stimulated by rapid changes in the thalamic firing rate, the intracortical circuits are low-pass and respond most strongly to slowly varying inputs from the cortical layer-4 population.     

Shrinkage and Pretest Nonparametric Estimation of Regression Parameters from Censored Data with Multiple Observations at Each Level of Covariate

A simple linear regression model is considered where the independent variable assumes only a finite number of values and the response variable is randomly right censored. However, the censoring distribution may depend on the covariate values. A class of noniterative estimators for the slope parameter, namely, the noniterative unrestricted estimator, noniterative restricted estimator and noniterative improved pretest estimator are proposed. The asymptotic bias and mean squared errors of the proposed estimators are derived and compared. The relative dominance picture of the estimators is investigated. A simulation study is also performed to asses the properties of the various estimators for small samples.     

Model selection and estimation in the Gaussian graphical model

We propose penalized likelihood methods for estimating the concentrationmatrix in the Gaussian graphical model. The methods lead toa sparse and shrinkage estimator of the concentration matrixthat is positive definite, and thus conduct model selectionand estimation simultaneously. The implementation of the methodsis nontrivial because of the positive definite constraint onthe concentration matrix, but we show that the computation canbe done effectively by taking advantage of the efficient maxdetalgorithm developed in convex optimization. We propose a BIC-typecriterion for the selection of the tuning parameter in the penalizedlikelihood methods. The connection between our methods and existingmethods is illustrated. Simulations and real examples demonstratethe competitive performance of the new methods.     

On the Estimation of Relative Risks and Longevities of a Class of Cancer Patients Based on Randomly Censored Data

An attempt has been made to measure the relative risks and longevities of a group of cancer patients using a Weibull model whose parameters are function of covariatcs (viz treatment and condition of the patient). The methodology is Cox's Partial Likelihood and the Maximum Likelihood techniques for randomly censored data. The data consists of the records of 134 patients under a clinical trial in the treatment of Carcinoma of the Oro Pharynx as quoted by Kalbfleisch and Prentice (1980).     

Coarse Grained Normal Mode Analysis vs. Refined Gaussian Network Model for Protein Residue-Level Structural Fluctuations

We investigate several approaches to coarse grained normal mode analysis on protein residual-level structural fluctuations by choosing different ways of representing the residues and the forces among them. Single-atom representations using the backbone atoms C _α , C, N, and C _β are considered. Combinations of some of these atoms are also tested. The force constants between the representative atoms are extracted from the Hessian matrix of the energy function and served as the force constants between the corresponding residues. The residue mean-square-fluctuations and their correlations with the experimental B-factors are calculated for a large set of proteins. The results are compared with all-atom normal mode analysis and the residue-level Gaussian Network Model. The coarse-grained methods perform more efficiently than all-atom normal mode analysis, while their B-factor correlations are also higher. Their B-factor correlations are comparable with those estimated by the Gaussian Network Model and in many cases better. The extracted force constants are surveyed for different pairs of residues with different numbers of separation residues in sequence. The statistical averages are used to build a refined Gaussian Network Model, which is able to predict residue-level structural fluctuations significantly better than the conventional Gaussian Network Model in many test cases.     

Predicting Important Residues and Interaction Pathways in Proteins Using Gaussian Network Model: Binding and Stability of HLA Proteins

A statistical thermodynamics approach is proposed to determine structurally and functionally important residues in native proteins that are involved in energy exchange with a ligand and other residues along an interaction pathway. The structure-function relationships, ligand binding and allosteric activities of ten structures of HLA Class I proteins of the immune system are studied by the Gaussian Network Model. Five of these models are associated with inflammatory rheumatic disease and the remaining five are properly functioning. In the Gaussian Network Model, the protein structures are modeled as an elastic network where the inter-residue interactions are harmonic. Important residues and the interaction pathways in the proteins are identified by focusing on the largest eigenvalue of the residue interaction matrix. Predicted important residues match those known from previous experimental and clinical work. Graph perturbation is used to determine the response of the important residues along the interaction pathway. Differences in response patterns of the two sets of proteins are identified and their relations to disease are discussed.